Homogenization of Hamilton-Jacobi Equations and Related Inverse Type Problems

Abstract

The speaker will first present some recent results on the homogenization theory for Hamilton-Jacobi equations in dynamic random environments. The goal of homogenization is to determine some effective environment that is non-oscillatory but characterizes the averaged effect of the heterogeneous media on the equation. Though the homogenization theory for Hamilton-Jacobi equation is well studied for static environments, some difficulty persists in the dynamic setting due to the lack of uniform Lipschitz controls of the solutions. The presented results, albeit being partial, provide new unified approaches for qualitative stochastic homogenization.

For the second part of the talk, the speaker will report some new studies on finer properties of the effective Hamiltonian, and some information about the environment that can be deduced from the effective Hamiltonian. This is, in some sense, inverse type problems and the speaker will show some examples in the periodic setting. This talk is based on joint works with P.E. Souganidis, H.V. Tran and Y. Yu.

About the speaker

Prof. Jing Wenjia received his BS from Peking University in 2006 and PhD from Columbia University in 2011. He then joined the École Normale Supérieure at Paris as a postdoctoral researcher from 2011 to 2013 and was a Dickson Instructor of Mathematics at the University of Chicago from 2013 to 2016. He has moved to Tsinghua University and is currently an Assistant Professor at the Yau Mathematical Sciences Center.

Prof. Jing works on partial differential equations with random coefficients, especially homogenization theory, waves in random media and their applications in imaging and other inverse problems.

About the program

For more information, please refer to the program website http://iasprogram.ust.hk/inverseproblems for details.